WO2016197629A1

WO2016197629A1 - System and method for frequency estimation

Info

Publication number: WO2016197629A1
Application number: PCT/CN2016/074688
Authority: WO
Inventors: Hongqing LIU; Yong Li; Yi Zhou
Original assignee: Chongqing University Of Posts And Telecommunications
Priority date: 2015-06-10
Filing date: 2016-02-26
Publication date: 2016-12-15
Also published as: CN104954298B; CN104954298A

Abstract

A method for reconstructing a signal is provided. The method may include determining a first function indicative of the signal, and initializing a missing pattern relating to the first function, under the impulsive noise condition. Noise reduction can be performed by solving an estimation problem utilizing the sparse structure of the noise in the time domain. An iterative estimation of frequency and missing pattern in the case of the impulsive noise may be conducted as follows: during each of a plurality of iterations, conducting a frequency and a noise estimation of the signal at each successive iteration based on the initialized or a prior calculated missing pattern, and conducting a missing pattern estimation based on the estimated frequency and noise and the first function generating at least one of a reconstructing the signal based on the estimated frequency, the estimated missing pattern, and the estimated noise.

Description

SYSTEM AND METHOD FOR FREQUENCY ESTIMATION

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority of Chinese Patent Application No. 201510315764.4 filed June 10, 2015, the entire contents of which are incorporated herein by reference.

TECHNICAL FIELD

The present disclosure relates to signal processing. More particularly, the present disclosure relates to a system and method for signal reconstruction by performing frequency estimation.

BACKGROUND

Frequency estimation is an important process in communication, navigation, radar and various other engineering systems. In some cases, the ability to perform efficient and precise estimation of frequency may be critical to system design. Inefficient and/or inaccurate estimation may significantly limit the performance of these systems as measured by various metrics of performance. In some communication systems, a signal is modulated to a carrier signal and transmitted over a communication channel. The carrier signal is modulated by varying one or more of its parameters (e.g., the amplitude, the frequency, or the phase of the carrier signal) according to information being transmitted. During transmission, the communication channel may alter the characteristics of the transmitted signal and may add random interference to the transmitted signal (e.g., Additive White Gaussian Noise (AWGN) ) . This may deteriorate the system performance (e.g., by making the transmitted signal difficult to be recovered at a receiver) .

Another issue encountered in communication is missing measurements due to data transmission loss, sensors’ failure, or other unknown reasons. Some useful signal samples being lost during signal transmission may cause problems, such as failure to recognize the information by the user.

SUMMARY

In accordance with some embodiments of the disclosed subject matter, methods, systems and media for processing a signal are provided.

In accordance with some embodiments of the disclosed subject matter, a method for processing a signal is provided. The method may include: constructing a signal model representative of the signal； determining an initial missing pattern vector relating to the signal model； conducting, by a processor, a frequency estimation on the signal based on the initial missing pattern vector； estimating a missing pattern based on the frequency estimation； and reconstructing the signal based on the frequency estimation and the missing pattern.

In some embodiments, conducting a frequency estimation may include estimating a first sparse vector representative of sparsity of the signal and at least one component of the first sparse vector corresponds to a frequency components of the signal.

In some embodiments, conducting a noise estimation may include estimating a second sparse vector based on the first sparse vector, and the second sparse vector corresponds to impulsive noise.

In some embodiments, estimating the missing pattern may include estimating an updated missing pattern vector based on the first sparse vector and estimating a third sparse vector.

In some embodiments, the missing pattern may correspond to a portion of an image.

In some embodiments, at least a component of the initial missing pattern vector may be zero.

In some embodiments, the signal model may include a superposition of multiple sinusoids, and each of the sinusoids corresponds to a frequency component of the signal.

In accordance with some embodiments of the disclosed subject matter, a system for processing a signal may be provided. The system may include at least one hardware processor. The hardware processor may be configured to: construct a signal model representative of the signal； determine an initial missing pattern vector relating to the signal model； conduct a frequency estimation on the signal based on the initial missing pattern vector； estimate a missing pattern based on the frequency estimation； and reconstruct the signal based on the frequency estimation and the missing pattern.

In some embodiments, to conduct the frequency estimation, the hardware processor is further to estimate a first spare vector representative of sparsity of the signal and at least one component of the first sparse vector corresponds to a frequency component of the signal.

In some embodiments, the hardware processor is further configured to estimate the missing pattern comprising estimating an updated missing pattern vector based on the first sparse vector.

In some embodiments, the hardware processor is further configured to conduct a noise estimation based on the first sparse vector. In some embodiments, conducting a noise estimation may include estimating a second sparse vector based on the first sparse vector, and the second sparse vector corresponds to impulsive noise.

In some embodiments, to conduct the missing pattern estimation, the hardware processor is further to estimate a third sparse vector.

In accordance with some embodiments of the disclosed subject matter, a non-transitory media containing computer-executable instructions that, when executed by a processor, may cause the processor to perform a method for processing a signal are provided. In some embodiments, the method may include the following steps: constructing a signal model representative of the signal； determining an initial missing pattern vector relating to the signal model； conducting a frequency estimation on the signal based on the initial missing pattern vector； estimating a missing pattern based on the frequency estimation； and reconstructing the signal based on the frequency estimation and the missing pattern.

BRIEF DESCRIPTION OF THE DRAWINGS

Various exemplary embodiments of the disclosed systems and methods for implementing signal processing in data communications and/or other applications will be described, in detail, with reference to the following drawings, in which:

FIG. 1 illustrates a block diagram of an exemplary system for signal processing according to some embodiments of the present disclosure；

FIG. 2 depicts a block diagram of an exemplary parameter estimator according to some embodiments of the present disclosure.

FIG. 3 is a diagram illustrating an example of a noise signal according to some embodiments of the present disclosure ；

FIG. 4 depicts a noise remodeling according to some embodiments of the present disclosure；

FIG. 5 illustrates an example of data missing according to some embodiments of the present disclosure；

FIG. 6 depicts an exemplary flowchart of the signal processing according to some embodiments of the present disclosure；

FIG. 7 depicts another exemplary flowchart of the signal processing according to some embodiments of the present disclosure；

FIG. 8 depicts another exemplary flowchart of the signal processing according to some embodiments of the present disclosure；

FIG. 9 depicts another exemplary flowchart of the signal processing according to some embodiments of the present disclosure；

FIG. 10 illustrates the architecture of a computer that may be used to implement a specialized system or a part thereof incorporating the present disclosure；

FIG. 11 depicts an exemplary signal according to some embodiments of the present disclosure；

FIG. 12 depicts an exemplary reconstructed signal according to some embodiments of the present disclosure；

FIG. 13 depicts reconstructed signals by different methods according to some embodiments of the present disclosure； and

FIG. 14 depicts another exemplary reconstructed signal according to some embodiments of the present disclosure.

DETAILED DESCRIPTION

The following description is presented to enable any person skilled in the art to make and use the disclosure, and is provided in the context of a particular application and its requirements. Various modifications to the disclosed embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the spirit and scope of the present disclosure. Thus, the present disclosure is not limited to the embodiments shown, but is to be accorded the widest scope consistent with the claims.

It will be understood that when a module or unit is referred to as being “on” , “connected to” or “coupled to” another module or unit, it may be directly on, connected or coupled to the other module or unit or intervening module or unit may be present. In contrast, when a module or unit is referred to as being “directly on, ” “directly connected to” or “directly coupled to” another module or unit, there may be no intervening module or unit present. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items.

The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. As used herein, the singular forms “a” , “an” and “the” may be intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising, ” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

When constructing a signal including additive noise, conventional frequency estimation techniques performed frequency estimation under the assumption that the additive noise is Gaussian noise with a Gaussian distribution. However, the additive noise may include non-Gaussian noise, such as impulsive noise. As such, these techniques fail to provide solutions for impulsive noise reduction while performing frequency estimation. Additionally, the conventional approaches fail to provide mechanisms for estimating arbitrary missing patterns while performing frequency estimation and/or signal reconstruction. For example, the conventional techniques interpolated a missing data point based on known data points (e.g., previously gathered data samples) . Thus, the conventional techniques cannot provide accurate estimation of a large amount of missing data and/or consecutive missing data.

To address these and other deficiencies of the conventional techniques, implementations of the present disclosure provide for mechanisms (which can include methods, systems, computer-readable medium, etc. ) for frequency estimation, missing pattern estimation, and impulsive noise reduction. For example, the mechanisms can perform frequency estimation on a signal based on sparsity of the signal. In some embodiments, the sparsity of the signal may be measured by a sparse vector that represents frequency sparsity of a representation of the signal in the frequency domain. As another example, the mechanisms can estimate a missing pattern based on sparsity of the signal (e.g., by constructing a sparse vector) . As still another example, the mechanisms can perform impulsive noise reduction on the signal by estimating an unknown sparse vector representative of the impulsive nature of one or more noise components of the signal.

The frequency estimation, the noise estimation, and/or the missing pattern estimation may be jointly carried out. For example, a joint estimation may be performed to recover the frequency and missing pattern under the impulsive noise condition. As another example, a joint estimation may be an estimation of the frequency in the case when the missing pattern is taken into account. As still another example, a joint estimation may be an estimation of the missing pattern under the impulsive noise condition.

FIG. 1 illustrates a block diagram of an example 100 of a system for signal processing according to some embodiments of the present disclosure. The system 100 may include a user interface 110 via which a user may interact with the system 100, initiate operations, and/or control the process for signal processing. It shall be noted that the process of signal processing may be initiated by operation of one or more transceivers 170 that may be included in the system 100, or with which the system 100 may be associated. The user interface 110 may be configured as one or more mechanisms that can receive and/or process information provided by a user and/or perform any other function. Exemplary input user interface may include, a wireless or wired connected keyboard, a touchscreen with “virtual” buttons for communicating commands and information to the system 100, a microphone through which a user may provide oral commands to be “translated” by a voice recognition program, an image scanning device through which a user may provide identification information or gestures to be recognized as instructions to the system 100, the like, or a combination thereof. The user interface 110 may also include any other like device for operation of, and data exchange with, the system 100. One or more user inputs may be received via the user interface 110 to initiate one or more operations by one or more portions of the system 100.

It shall be noted that the above description of the user interface 110 is provided for the purpose of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, the variations and modifications may not depart the protecting scope of the present disclosure. For example, the user interface 110 may not be necessary. The initialization of the signal processing may be conducted under a predetermined condition, for example, an application program being executed, a certain kind of signal being detected, or the like, or a combination thereof.

The system 100 may include one or more processors 120 for undertaking the processing and control functions that are carried out by the system 100. The processors 120 may be local processor, or remote processors shared within a cloud. The processor (s) 120 may include one or more processors and/or microprocessor that can interpret and execute instructions and can process outgoing and incoming signals via, for example, the transceiver 170, or other communication links (not shown in FIG. 1) , to facilitate the signal processing algorithm described elsewhere in the present disclosure.

Additionally, the system 100 may include one or more data storage devices 130. The data storage device (s) 130 may be used to store data, operation programs, instructions, or applications to be used by the exemplary system 100, and specifically the processor (s) 120. The data storage device (s) 130 may include a random access memory (RAM) or another type of dynamic storage device that stores information and/or instructions for execution by the processor (s) 120. The data storage device (s) 130 may also include a read-only memory (ROM) , e.g., a ROM device or another type of static storage device that stores static information and/or instructions for execution by the processor (s) 120. Further, the data storage device (s) 130 may include a magnetic disk, an optical disc, a flash memory device, and/or a solid state storage device. In some examples, the data storage device (s) 130 may be integral to the system 100. In some examples, the data storage device (s) 130 may include a remote data storage device external to the system that is in wireless communication with the system 100, or removably connectable to the system 100 (e.g., a USB port, a firewire port, etc. ) . The data storage device (s) 130 may include or be connectively operational with one or more virtual storage resources (e.g., cloud storage, a virtual private network, and/or other virtual storage resources) . At least one of the data storage device (s) 130 may be used to store information relating to the signal processing, such as, signal processing algorithm, signal processing parameters, etc.

The system 100 may include at least one display device 140 that may be configured to display information relating to the signal processing to a user. The information relating to the signal processing may include, for example, various operating modes, a plurality of signal models, usable results on a resolved parameter estimation, recovered signal after processing, or the like, or a combination thereof.

The system 100 may include a communication interface 150 by which communication among different component elements of the system 100 are integral or not integral to a single device.

The system 100 may include a parameter estimator 160 that is configured to estimate relative parameters of the signal. Exemplary parameter estimations may include frequency estimation, missing pattern estimation, noise estimation, or the like, or a combination thereof. Different estimations may be conducted independently, or in a joint way. For examples, frequency estimation may be conducted independently by the parameter estimator 160. In some examples, frequency estimation, missing pattern estimation, and noise estimation may be conducted jointly. For instance, the frequency estimation and the missing pattern estimation may be conducted in an iterative way. Details of the estimation will be discussed elsewhere in the disclosure.

The system 100 may include at least one transceiver 170. The transceiver (s) 170 may be configured to receive the signal (s) to be processed. In some embodiments, the transceiver 170 may be a radio device that is configured to communicate real-time information, such as, a voice message, a video, an audio, with another device. For example, a voice message may be mixed with various kinds of noises that may come from noise of the communication channel or background noise. The noise of the communication channel may be in the form of Gaussian distribution, Cauchy distribution, Possion distribution, or impulsive noise distribution, or the like, or a combination thereof. The background noise may come from the sound of a speaker other than the user, the sound of different behaviors, such as keyboard tapping, coughing, etc. The background noise may also come from natural sounds, such as sound of thunder, sound of rain, etc. In some embodiments, the transceiver 170 may be configured to receive media information, such as a video, a picture, or the like, or a combination thereof. The media information may be received through a wired or wireless connection with a local storage device, e.g., the data storage device 130. Alternatively, the media information may be connected with a removable storage device or a remote storage device, such as a server, a cloud storage, or the like, or a combination thereof. The media information may relate to a picture with covered or blurring portions. Merely by way of example, a picture of a partially covered board may be communicated to the transceiver 170 for further processing in the system 100. In another example, image artifacts derived from image mosaic, or due to data loss may also be communicated to the transceiver 170 for further processing, such as image reconstruction, or image recovery, etc.

The system 100 may further include a signal recovery unit 180 that the signal (s) may be recovered based on the parameters estimated by the parameter estimator 160. In some embodiments, the parameter estimator 160 and the signal recovery unit 180 may be integrated in the processor 120.

The various components of the system 100, as depicted in FIG. 1, may be connected by one or more data/control busses. The data/control bus (es) may provide wired or wireless communication between the various components of the system 100. All of the components may be housed integrally, or may be housed separately and in wired or wireless communication with the system 100.

In some embodiments, each of user interface 110, processor 120, data storage device 130, display device 140, communication interface 150, and transceiver 170 may include and/or be any of a general purpose device such as a computer or a special purpose device such as a client, a server, and/or any other suitable device. Any of these general or special purpose devices may include any suitable components such as a hardware processor (which may be a microprocessor, digital signal processor, a controller, and/or any other suitable hardware processor) , memory, communication interfaces, display controllers, input devices, and/or any other suitable components. For example, each of user interface 110, processor 120, data storage device 130, display device 140, communication interface 150, and transceiver 170 may be implemented as or include a personal computer, a tablet computer, a wearable computer, a multimedia terminal, a mobile telephone, a gaming device, a set-top box, a television, and/or any other suitable device. Moreover, each of user interface 110, processor 120, data storage device 130, display device 140, communication interface 150, and transceiver 170 may include a storage device, which may include a hard drive, a solid state storage device, a removable storage device, and/or any other suitable storage device. Each of user interface 110, processor 120, data storage device 130, display device 140, communication interface 150, and transceiver 170 may be located at any suitable location. Each of user interface 110, processor 120, data storage device 130, display device 140, communication interface 150, and transceiver 170 may be implemented as a stand-alone device or integrated with other components of system 100.

It shall be noted that the above description of various components of the system 100 is provided for the purpose of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, the variations and modifications may not depart the protecting scope of the present disclosure. The various components of the system 100 may be arranged in combinations of sub-systems as individual components or combinations of components. For instance, the parameter estimator 160 and signal recovery unit 180 may be integral to a single unit to perform the signal processing.

Referring to FIG. 2, the parameter estimator 160 may include a frequency estimation module 210, a noise estimation module 220, a missing pattern estimation module 230, an estimation controller 240, and/or any other suitable component for performing frequency estimation in accordance with some embodiments of the present disclosure. In some embodiments, a signal may be processed by one or more of the frequency estimation module 210, the noise estimation module 220, and/or the missing pattern estimation module 230. The signal may be processed under the control of the estimation controller 240 based on, for example, the commands input through the user interface 110, the programs or instructions stored in the data storage device 130, the control functions carried out by the processor 120, the like, or a combination thereof.

The frequency estimation module 210 may be configured to carry out frequency estimation on a signal received by the parameter estimator 160. The received signal may be a continuous signal, a discrete signal, etc. The received signal may include information about any content, such as video content, audio content, text, images, etc. The received signal may be and/or include a speech signal, an image, a video, etc.

In some embodiments, the received signal may include a number of data samples. Each of the data samples may include one or more values at a point in time and/or space. Alternatively or additionally, one or more data samples may be extracted from the received signal using a sampler or any other signal processor.

The received signal may be processed by frequency estimation module 210, noise estimation module 220, missing pattern estimation module 230, and/or any other device as will be discussed in more detail below. Alternatively or additionally, the received signal can be divided into multiple blocks of data samples and/or subblocks of data samples. Each of the blocks of data samples and the subblocks of data samples may include any suitable number of data samples. Multiple blocks and/or subblocks may or may not overlap with each other. Each of the blocks of data samples and the subblocks of the data samples can be processed in accordance with various embodiments of the present disclosure. Multiple blocks and/or subblocks can be processed in parallel, sequentially, or in any other suitable manner.

In some embodiments, the frequency estimation module 210 can construct a signal model representing the received signal. For example, the received signal may be represented as a combination of a desired signal component, a noise component, and/or any other component. More particularly, for example, the received signal may be represented as follows:

z(k) ＝x (k) +n (k) , k＝0,…, N-1. (1)

In equation (1) , z (k) may denote the received signal for further processing. x (k) may denote a noise-free signal. n (k) may denote an additive noise signal (e.g., a Gaussian noise, a Cauchy noise, a Poisson noise, an impulsive noise, etc. ) . k may represent an index of data samples of the received signal. N may represent the length of received signal (e.g., the number of the data samples of the received signal) . As used herein, the noise signal may represent the “unwanted signal” that appears along with the noise-free signal, or be generated during the transmission and/or processing of the noise-free signal.

In some embodiments, the received signal may be sparse in a time domain. For example, one or more portions of the received signal (e.g., the noise signal, the desired signal, and/or any other component of the received signal) may be sparse in the time domain. In some embodiments, the received signal may be sparse in a transform domain (e.g., a frequency domain) . For example, one or more portions of the received signal (e.g., the noise signal, the desired signal, etc. ) may be sparse in the frequency domain. More particularly, for example, a sparse noise signal n (k) may be represented as a combination of one or more non-zero components and one or more zero components. In some embodiments, one or more portions of the received signal (e.g., the noise signal) does not appear consecutively in the time domain, or in any transform domain (e.g., the frequency domain) .

In a more particular example, a frequency estimation may be performed based on the signal sparsity of the received signal or part of the received signal in the frequency domain. For example, considering the one-dimensional (1-D) frequency estimation, the desired signal may be depicted as a superposition of L sinusoids according to the following equation:

where ω_l∈ [0, 2π) may be a frequency component of the received signal (e.g., the lth frequency component) , and a_l may be the amplitude of the frequency component. In equation (2) , the desired signal represented by the superposition of L sinusoids may indicate that the desired signal is sparse and/or can be represented in a sparse form in the frequency domain. In some embodiments, the received signal is not sparse in the time domain and may cover the whole time indexes. The transform domain of the received signal may be utilized to determine sparsity of the received signal. In some embodiments, the frequency estimation module 210 may generate a representation of the received signal in the frequency domain by performing, for example, a time-frequency transformation on the received signal. For example, the spectrum of the received signal may be generated by performing the Fourier transform (FT) , the short-time Fourier transform, the wavelet transform, the Wigner distribution function (WDF) , the modified Wigner distribution function, the Gabor-Wigner distribution function, etc., on the received signal. In some embodiments, a frequency representation of the received signal may be expressed as:

where δ (·) denotes the Dirac delta function. In some embodiments, the received signal may be L-sparse in the frequency domain since δ (·) is an impulse function. For example, the L frequency components denoted by ωl are non-zero while the other frequency components of the received signal are zero. The received signal (e.g., the signal expressed as in equation (2)) may be represented in matrix form:

z＝Da+n, (4)

where z＝ [z₁, …, z_N] ^T denotes a matrix representing the received signal, a＝[a₁, …, a_L] ^T denotes a vector representing the amplitudes of the frequency components of the received signal, n＝ [n₁, …n_N] ^T denotes a matrix representing the noise signal, and D denotes a N×L matrix. In some embodiments, matrix D may be represented as:

The frequency estimation module 210 can perform frequency estimation on the received signal based on the sparsity of the received signal. For example, taking a grid spectrum over the range [0, 2π) into account, the unknown frequency components ωl may be overlapped on top of the gridding spectrum. Thus, the amplitudes of unknown frequency components (e.g., frequency components that are included in the received signal) may have non-zero values. The amplitudes of the other frequency components (e.g., frequency components that are not included in the received signal) may be zero. The non-zero values may be identical, or not identical based on the corresponding frequency components.

In some embodiments, the frequency estimation module 210 may divide the frequency range into a certain number of points. For example, the frequency range may be uniformly divided into and/or gridded by J frequency points. In such an example, a frequency point within the frequency range may be represented as f_j＝2jπ/J, where j＝0, …J-1. As another example, the frequency range may be non-uniformly divided into several frequency points (e.g., based on a specific rule) . In some embodiments, a J-point DFT-like basis may be expressed as follows:

The frequency range may be gridded to any suitable number of frequency points. For example, the number of frequency points may be determined based on the value of L. In a more particular example, the value of J may be set to satisfy J ＞＞L to achieve a fine estimation. As another example, the number of the frequency points and/or the value of J may be set as a constant and may be used in frequency estimation on one or more received signals. As a further example, the number of the frequency points and/or the value of J may be a variable and may be adjusted based on the condition of the frequency estimation (e.g., the accuracy of the frequency estimation, etc. ) , a user input (e.g., an input received via the user interface 110 and/or any other user input) , the types of signals received (e.g., a voice signal, an image, and/or any other types of signals) , and/or any other information that may be used to determine the value of J. As still another example, the value of J may be adjusted based on data about frequency estimation performed on one or more signals. More particularly, for example, if the received signal is similar to one or more and/or one or more portions of other signals (e.g., one or more previously estimated signals) , the value of J may be adjusted based on the settings of previous frequency estimation performed on the similar signal (s) (e.g., by setting the value of J as the same as a number of frequency points used in the previous frequency estimation) . The data about the previous frequency estimation may be stored in the data storage device 130, a cloud-based storage device, a common shared medium, and/or any other storage device. In some embodiments, a previously estimated signal may be regarded as being similar to the received signal when the previously estimated signal and the received signal present one or more similar or same characteristics. For example, the previously estimated signal and the received signal may be received via similar communication path or channel. As another example, the previously estimated signal and the received signal may include similar noise signals, similar desired signals, and/or any other component.

In some embodiments, the frequency estimation module 210 can measure the sparsity of the received signal by determining a sparse vector including the coefficients corresponding to the frequencies in the frequency range. For example, the sparse vector may be represented as vector x＝ [x₁, …, x_j, …, x_J] ^T, where x_j is the coefficient corresponding to the jth point in the frequency range (e.g., f_j) . In some embodiments, a non-zero coefficient may indicate that a frequency component corresponding to the coefficient is included in the received signal. Similarly, a zero coefficient may indicate that a frequency component corresponding to the coefficient is not included in the received signal.

The value of a given coefficient of the sparse vector may indicate whether a frequency corresponding to the coefficient is on the grid f_j. As such, the sparse vector may indicate the sparsity of the received signal. For example, if a coefficient of the sparse vector is not zero, the frequency lying on the grid corresponding to the coefficient may exist (e.g., being included) in the received signal. As another example, if a coefficient of the sparse vector is zero, the frequency lying on the grid corresponding to the coefficient is not included in the received signal. In some embodiments, one or more coefficients in vector x may have a value of zero and may correspond to frequencies that are not included in the received signal. In some embodiments, the coefficients of the vector x may be determined based on the following equation:

The frequency estimation module 210 can perform frequency estimation on the received signal based on the sparse vector. In some embodiments, the sparse vector may be estimated by solving an optimization problem. One or more unknown frequencies of the received signal and their amplitudes may then be determined based on the estimated sparse vector.

For example, according to equation (6) , the frequency estimation may be performed by solving the following optimization problem:

minimize||z-Ψx||₂. (8)

As another example, the frequency estimation module 210 can perform a frequency estimation by solving the following optimization problem:

minimize ||z-Ψx||₂+μ||x||₁, (9)

where z denotes the received signal； ||·||₁ is l1 norm that may be utilized to enforce the solution of the vector x. In some embodiments, other norm l_p, (e.g., 0<p<1) may be used as well to enforce the solution. In some embodiments, a sparse vector x corresponding to sparse frequency components contained in the received signal may satisfy equation (9) and the frequencies corresponding to non-zero coefficients of the vector x on the grid f_j may indicate the unknown frequency components in the received signal.

The noise estimation module 220 may be configured to perform noise reduction on a signal received by the parameter estimator 160. For example, the noise reduction module 220 can estimate a noise component of the received signal and can subtract the estimated noise component from the received signal. The noise estimation module 220 can perform noise estimation and/or reduction individually, or in combination with other modules in the parameter estimator 160. In some embodiments, one or more portions of the noise component may be represented by the Gaussian distribution or a non-Gaussian distribution. Exemplary non-Gaussian noise may include at least a significant peak pulse waveform and/or thick tail probability density functions. In some embodiments, the type of noise may be modeled by α-stable distribution. For example, a probability density function (PDF) may be used to define the noise distribution as:

where

In equations (10) - (11) , α denotes the characteristic exponent that may determine the shape of the distribution and may control the heaviness of the tails of the density function. For example, the heavier the tails is, the more severe the impulsiveness may be. In some embodiments, the value of α may be adjusted in a range (e.g., (0<α≤2) or any other suitable range) . γ denotes the dispersion parameter that may determine the spread of the density and may act in a similar way to the variance of the Gaussian density. In some embodiments, the value of γ may be adjusted in a range (γ>0) . In some embodiments, the signal noise rate (SNR) may be defined by the characteristic exponent α and the dispersion parameter γ as:

where P_sig may be the signal energy. β (-1≤β≤1) denotes a symmetry parameter, and β＝0 may correspond to the symmetric α-stable (SαS) distribution, i.e., symmetric about a.

(iv) a (-∞<a<∞) denotes the location parameter that may be the mean value when 1<a≤2, and the median value when 0<a<1 for SαS distributions. Different values of α and β may correspond to different pulse waveforms. For example, the α-stable distribution may be simplified to Gaussian distribution when α＝2, and converted to Cauchy distribution when α＝2, β＝0.

In some embodiments, the noise component of the received signal may represent an impulse noise. The noise signal may be modeled as a sparse or nearly-sparse signal. For example, as illustrated in diagram 300 of FIG. 3, a noise signal 310 may include one or more spikes representing impulsive noise components of noise signal 310 (e.g., sparks 311 and 313) and one or more frequency components representing residual noises of noise signal 300 (e.g., values 305) . As shown, the x axis of diagram 300 represents the sample number of noise signal 310. The y axis of the diagram shows the amplitudes corresponding to various frequency components and/or sample numbers of noise signal 310. One or more spikes representative of impulsive noise components of noise signal 310 (e.g., sparks 311 and 313) may have an amplitude greater than a threshold. One or more amplitudes of frequency components corresponding to residual noise components (the small values 315) may be less than a threshold. As used herein, a spike may denote an amplitude of a frequency component that is greater than a threshold and/or one or more frequency components. For example, the amplitude of a spike may be equal to or significantly greater than one or more other frequency components (e.g., 5 times greater, 10 times greater, 20 time greater, etc. ) . The spikes and the small values may be formulated by different distributions to describe the noise signal. For example, the big spikes of the noise and the small value noise may be represented using a sparse vector and a residual vector, respectively. The noise signal may be modeled as a combination of the sparse vector and the residual vector. More particularly, for example, the noise signal may be modeled as follows:

n＝e+w, (14)

where n, e, and w represent the noise signal, the sparse vector, and the residual vector, respectively.

For example, as illustrated in FIG. 4, a noise signal 410 may be represented as a combination of a sparse vector 420 and a residual vector 430. In some embodiments, residual vector 430 may be and/or include a Gaussian vector. The sparse vector 420 may model the big spikes illustrated in FIG. 3. The residual vector 430 may model a residual noise (e.g., a Gaussian noise) illustrated in FIG. 3.

Referring back to FIG. 2, the noise estimation module 220 can estimate the impulsive noise by determining the sparsity of the noise signal (e.g., in the time domain) . For example, the estimation may be performed by solving an optimization problem, such as the following optimization problem:

minimize ||n||₁

subject to z ＝Da +n. (15)

The value of vector n may indicate when a noise component appears in the received signal. As such, the sparse vector n may indicate the impulsive property of the noise in the received signal. In some embodiments, the noise component may be estimated by taking residue into account:

minimize ||n||₁

subject to ||z-Da-n||₁<ε. (16)

Accordingly, the sparse noise estimation may be realized by estimating a value of the noise component (e.g., a combination of the sparse vector and the residual vector) to minimize a norm of the noise component (e.g., ||n||₁) . By performing an estimation based on equation (16) , a small variance in the noise signal (e.g., due to the nearly-sparse property) may be taken into account by the noise estimation module 220.

The missing pattern estimation module 230 may be configured to estimate patterns of missing data in the received signal (also referred to herein as the “missing pattern” ) . The missing patterns may be caused by, for example, sensors’ failure, data transmission loss, and/or any other reason. In some embodiments, data missing may occur during a wireless or wired communication including real-time communication, e.g., communication by voice message, audio communication, video communication, the like, or a combination thereof. For instance, data missing may relate to packets loss or collision, or part of the useful information submerged by environmental noise, or the like, or a combination thereof. Data missing during the wireless or wired communication may be caused by certain wireless or wired communication channels. In some embodiments, data missing may occur in the form of media information, including a picture, a frame of a video, or the like, or a combination thereof. For instance, data missing may relate to covered or blurring portions in the picture. Data missing of media information, such as a picture, may be caused by sheltering, smearing, coverings objects of interest intentionally or accidentally. In some embodiments, data missing may occur at times when signals relating to environmental measurements are detected by sensors. Exemplary environmental measurements may include temperature, light, sound, humidity, the like, or a combination thereof. Data missing may relate to low sensor battery levels, potential harsh environmental, the like, or a combination thereof.

The missing pattern estimation module 230 may sample the received signal to obtain one or more data samples of the received signal. Alternatively or additionally, the missing pattern estimation module 230 may receive the data samples from any other device. The received signal may be sampled at any suitable sampling rate. The data samples may be obtained by sampling the received signal using one or more uniform sampling techniques, non-uniform sampling techniques, and/or any other suitable sampling technique. For example, as illustrated in FIG. 5. As shown in diagram 510, regular uniform data samples may be gathered. As another example, as shown in diagram 520, a random missing pattern may be produced that a portion of the data samples may not be acquired at, for example, the transceiver 170.

In some embodiments, missing pattern estimation may be performed based on a sparse vector. The missing data samples may correspond to zero components in the sparse vector. For example, if a signal transmitted to system 100 including uniform samples [y₁, y₂, y₃, y₄] ^T, the signal received at the transceiver 170 and/or the missing pattern estimation module 230 may include fewer data samples due to various reasons. For example, data samples of y₂ and y₃ may be missing during transmission. As such, the received signal may include data samples of y₁ and y₄. The received signal may be modeled as [y₁, y₄] ^T＝s。 [y₁, y₂, y₃, y₄] ^T, where s＝ [1, 0, 0, 1] ^T and 。is Schur product to indicate componentwise product. The value of a coefficient of vector s may indicate whether a data sample is missing. The frequency estimation signal model with data missing may be expressed as:

where

denotes the received signal with missing measurements and s denotes a sparse vector representative of a missing pattern. Suppose the frequency vector a is known (e.g., determined by the frequency estimation module 210 and/or any other device) , missing data estimation may be performed by estimating sparse vector s. For example, missing data estimation may be performed by solving the following convex optimization:

minimize ||s||₁

Thus, the missing pattern estimate may be realized by minimizing a norm of the sparse vector (e.g., ||s||₁) .

The estimation controller 240 may be configured to control the frequency estimation module 210, the noise estimation module 220, and/or the missing pattern estimation module to perform frequency estimation, missing pattern estimation, and/or noise estimation and reduction. In some embodiments, the frequency estimation module 210, the noise estimation module 220, or the missing pattern estimation module 230 may carry out frequency estimation, noise estimation, and missing pattern estimation, respectively. Alternatively or additionally, one or more of these modules can perform joint frequency, noise, and/or data missing patter estimation. For example, one or more operations described in conjunction with FIGS. 6-9 may be performed by one or more of the frequency estimation module 210, the noise estimation module 220, and/or the missing pattern estimation module 230.

In some embodiments, the frequency estimation, the noise estimation, and/or the missing pattern estimation may be jointly carried out under the control of the estimation controller 240. For example, a joint estimation may include simultaneously recovering the frequency and missing pattern under the impulsive noise condition. In another example, a joint estimation may be an iterative estimation of the frequency and the missing pattern. In another example, a joint estimation may be the frequency estimation under the impulsive noise condition. In still another example, a joint estimation may be the estimation of the missing pattern under the impulsive noise condition. The type of the joint estimation may be controlled by a user input through the user interface 110, or may follow a preset schedule stored in the data storage device 130 or executed by the processor 120.

FIG. 6 depicts a flowchart illustrates an example 600 of a process for signal processing according to some embodiments of the present disclosure. The method 600 may be performed by processing logic that comprises hardware (e.g., circuitry, dedicated logic, programmable logic, microcode, etc. ) , software (e.g., instructions run on a processing device to perform hardware simulation) , or a combination thereof. In some embodiments, one or more operations of the method 600 can be performed by one or more computing devices (e.g., one or more computing devices as described below in connection with FIG. 10) executing parameter estimator 160 and/or signal recovery unit 180 of FIG. 1.

At step 610, a signal model may be established. The signal model may relate to the type of estimation (s) to be performed in the following steps, such as frequency estimation, missing pattern estimation, and/or noise estimation, etc. Exemplary signal models may be stored in the data storage device 130, or may be revised or adjusted by a user through the user interface 110.

At step 620, a signal may be received. In some embodiments, the received signal may include video content, audio content, images, text, and/or any other media content. The received signal can include real-time communication information and can be and/or include a voice message, a video, an audio, the like, or a combination thereof. The signal may be received via one or more sensors, transceivers (e.g., a transceiver 170 of FIG. 1) , and/or any other device that is capable of receiving a signal. The signal may be transmitted via any suitable communication channel. In some embodiments, the transmitting environment of signal may include an open space, shallow waters, cables, or the like, or a combination thereof.

At step 630, a noise model, a missing pattern model, and/or a frequency estimation model may be established. For example, the frequency estimation may be calculated with a maximum likelihood (ML) method, a linear prediction approach, a joint estimation as described elsewhere in the disclosure, or the like, or a combination thereof. The missing pattern may be calculated by a gapped-data amplitude and phase estimation (GAPE) algorithm, an autoregressive moving average (ARMA) model, an iterative adaptive approach (IAA) , a joint estimation as described elsewhere in the disclosure, or the like, or a combination thereof. The noise model may be a Gaussian noise, a Cauchy noise, an impulsive noise, or other stable distribution noise. In some embodiments, the noise model, missing pattern model, and/or frequency estimation model may be established by performing one or more operations described in conjunction with FIGS. 7-9 below.

At step 640, estimation of frequency, missing pattern, and/or noise may be performed. One or more frequency estimation, missing patter estimation, and noise estimation may be performed in parallel, sequentially, and/or in any other suitable manner. In some embodiments, a joint estimation of frequency, missing pattern, and/or noise can be performed (e.g., by performing one or more operations described in connection with FIGS. 7-9 below.

At step 650, the signal may be reconstructed based on the estimation (s) to generate a reconstructed signal. For example, missing data samples of the received signal may be recovered based on the missing pattern estimation. As another example, one or more frequency components and/or their amplitudes may be recovered based on the frequency estimation. As still another example, estimated noise components (e.g., noise components corresponding to impulsive noise, Gaussian noise, etc. ) may be subtracted from the received signal to generate a denoised signal.

It shall be noted that the above description of the flowchart of signal processing is provided for the purpose of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, the variations and modifications may not depart the protecting scope of the present disclosure. The flow diagrams may be executed or performed in any order or sequence not limited to the order and sequence shown and described in the figures. For example, the step 620 may occur in advance to the step 610, i.e., a signal model may be established after a signal is received. Thus, the signal model may be selectively established based on the received signal. In another example, if the signal is not well recovered or a user may want to achieve a more accurate recovery, it may loop back to step 610, or step 630 to re-establish a noise model, a missing pattern model, and/or a frequency estimation model. In a further example, more types of estimations may be carried out during the signal processing. The types of estimations may relate to the phase shift, amplitude change, cross-correlation between signals, autocorrelation of signals, or the like, or a combination thereof.

FIG. 7 depicts a flowchart illustrating an example 700 of a process for performing joint frequency and noise estimation according to some embodiments of the present disclosure. The method 700 may be performed by processing logic that comprises hardware (e.g., circuitry, dedicated logic, programmable logic, microcode, etc. ) , software (e.g., instructions run on a processing device to perform hardware simulation) , or a combination thereof. In some embodiments, one or more operations of the method 700 can be performed by one or more computing devices (e.g., one or more computing devices as described below in connection with FIG. 10) executing parameter estimator 160 and/or signal recovery unit 180 of FIG. 1.

At step 710, a signal model may be constructed. For example, the signal model may represent a signal as a combination of a desired signal component, a noise component, and/or any other component. The desired signal component may correspond to a speech signal, a video signal, and/or any other signal that may be regarded as being desirable. The noise component may represent Gaussian noise, impulsive noise, and/or any other type of noise. In some embodiments, the desired signal component may be represented as a superposition of multiple sinusoids. Each of the sinusoids may correspond to a component of the desired signal in the frequency domain. The different sinusoids may correspond to components with different frequencies of the desired signal in a frequency domain. In some embodiments, the desired signal component may be represented as superposition of multiple wavelets. Each of the wavelet may correspond to a component of the desired signal component in the frequency domain. The wavelets may correspond to frequency components of different portions of the desired signal. In some embodiments, the noise component may represent Gaussian noise, impulsive noise, and/or any other type of noise. The signal model may be constructed by performing one or more operations described in connection with equations (1) - (9) above.

In some embodiments, the signal model can be constructed by constructing a sparse vector representative of sparsity of the signal (also referred to herein as the “first sparse vector) . For example, the first sparse vector may represent sparsity of the signal in the frequency domain. The sparse vector can contain one or more coefficients corresponding to one or more frequency points. Each of the coefficients may correspond to an amplitude of its corresponding frequency point. The coefficients may be determined by performing one or more operations described in connection with equation (7) above.

At step 720, process 700 can construct a noise model. The noise model can represent one or more noise components of the signal modeled by the signal model. For example, the noise component (s) may be modeled as a combination of one or more components representative of various types of noises. In a more particular example, the noise component (s) may be modeled as a combination of a sparse vector (also referred to herein as the “second sparse vector” ) and a residual vector. The sparse vector can represent noise components with certain amplitudes (e.g., amplitudes greater than a threshold) . The residual vector can represent residual noise (e.g., noise components with amplitudes that are not greater than a threshold) . The residual vector may be a Gaussian vector or any other suitable vector that can represent residual noise components. In some embodiments, the second signal model can be constructed by performing one or more operations described in connection with equations 10-16 above.

At step 730, process 700 can receive a signal. For example, the signal may be received as described in conjunction with step 620 of FIG. 6.

At step 740, process 700 can perform joint frequency and noise estimation on the received signal based on the signal model and the noise model. The joint frequency and noise estimation may be performed to estimate frequency components of the received signal (e.g., frequencies and their amplitudes) and a noise component of the received signal (e.g., impulsive noise) . The estimated frequency components may be used to reconstruct a signal based on the received signal. The estimated noise component may be subtracted from the received signal to generate a denoised signal.

For example, the joint frequency and noise estimation may be performed by estimating the first sparse vector, the second sparse vector, and/or the residual vector. Values of the first sparse vector, the second sparse vector, and/or the residual vector may be estimated by solving one or more optimization problems. In some embodiments, a value of the first sparse vector and a value of the second sparse vector may be estimated to minimize combination of a norm of the first sparse vector and a norm of a combination of the second sparse vector and the residual vector. In a more particular example, the joint frequency and noise estimation may be performed by solving the following convex optimization problem:

minimize ||x||₁+τ||n||₁

subject to ||z-ψx-n||₁<ε, (19)

where vector x denotes the first sparse vector indicative of sparsity of the received signal； n denotes a noise component of the received signal； τ is a regularization constant； and ε is a small value. In some embodiments, vector x may be determined based on one or more operations described in connection with equations 1-8. Vector n may be determined based on one or more operations described in connection with equations 10-14.

It shall be noted that the above description of the flowchart of signal processing is provided for the purpose of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, the variations and modifications may not depart the protecting scope of the present disclosure. For example, the flow diagrams may be executed or performed in any order or sequence not limited to the order and sequence shown and described in the figures. A signal model may be established when, or after a signal is received. The signal model may further be selectively established based on the received signal. In another example, the received signal may relate to media information including a picture, a frame of a video, or the like, or a combination thereof. In some embodiments, the media information may be modeled in the same manner with the real-time communication information. For example, part of the media information, for example, the noise-free part, may be represented as superposition of multiple sinusoids, and part of the media information, for example, the part relating to noise, may be represented in the form of Gaussian distribution or impulsive noise distribution. In some embodiments, the media information may be modeled in a different way from the real-time communication information. For example, part of the media information, for example, the noise-free part, may be represented as superposition of multiple wavelets, and part of the media information, for example, the part relating to noise, may be represented in the form of Gaussian distribution, or impulsive noise distribution. In some embodiments, part of the media information may be represented as superposition of different types of forms, including sinusoids, wavelets, etc. The noise of the media information may relate to covered or blurring portions in the picture, artifacts derived from image/video mosaic, or data loss in a media information, or the like, or a combination thereof.

FIG. 8 depicts a flowchart illustrating an example 800 of a process for performing joint frequency estimation and missing pattern estimation according to some embodiments of the present disclosure. The method 800 may be performed by processing logic that comprises hardware, software, or a combination thereof. In some embodiments, one or more operations of the method 800 can be performed by one or more computing devices (e.g., one or more computing devices as described below in connection with FIG. 10) executing parameter estimator 160 and/or signal recovery unit 180 of FIG. 1.

At step 810, a signal model may be constructed. For example, the signal model may represent a signal with one or more missing data samples. The signal may be modeled based on a signal without missing data and a missing pattern vector representative of one or more missing patterns (also referred to herein as the “third sparse vector” ) . For example, the signal may be modeled as a Schur product of the signal without missing data and the third sparse vector. In some embodiments, the signal model may be constructed based on equation (17) . The signal without missing data may be modeled as a combination of a desired signal component, a noise component, and/or any other component. In some embodiments, the signal without missing data may be modeled using a signal model constructed at 710 of FIG. 7 above.

At step 820, process 800 can receive a signal. For example, the signal may be received as described in conjunction with step 620 of FIG. 6.

At step 830, process 800 can perform joint estimation of frequency and missing pattern based on the signal model. The frequency estimation may be performed to calculate existing frequency components of the received signal (e.g., frequencies and their amplitudes) . The missing pattern estimation may be performed to calculate the missing data samples of the received signal. For example, the joint frequency and missing pattern estimation may be performed by estimating the first sparse vector and the third sparse vector. The first sparse vector and/or the third sparse vector may be estimated by solving one or more optimization problems. For example, the first sparse vector and the third sparse vector may be estimated to minimize a combination of a norm of the first sparse vector and a norm of the third sparse vector. In a more particular example, the joint frequency and missing pattern estimation may be performed by solving the following optimization problem:

minimize ||x||₁ + τ||s||₁

subject to ||y-sο (ψx) ||₂<ε, (20)

where vector x denotes the first sparse vector indicative of sparsity of the received signal； s denotes the third sparse vector representative of one or more missing patterns； τ is the regularization constant； and ε is a small parameter that may be predefined based on user inputs, training data, and/or any other information.

In some embodiments, the joint estimation may be performed by iteratively performing frequency estimation and missing pattern estimation. For example, during each iteration, the first sparse vector may be estimated to produce one or more frequency components of the received signal and/or their amplitudes. The missing pattern estimation may then be performed based on the estimated frequency components and/or amplitudes. In some embodiments, the joint estimation may be performed by solving a convex optimization problem with respect to sparse vector x and sparse vector s. More particularly, for example, the joint estimation may be performed by performing one or more operations described in connection with blocks 840-870.

At step 840, a missing pattern vector may be initialized. For example, an initial missing pattern vector may be determined. The initial missing pattern vector x may be determined based on data stored in a storage device (e.g., the data storage device 130 or any other storage device) , one or more user inputs, and/or any other information. In some embodiments, the initial missing pattern vector may be a random vector (e.g., a vector randomly selected by the processor 120 in the system 100 or any other device) . In some embodiments, the missing pattern vector may be a predetermined vector that is preset by a user or a device.

At step 850, frequency estimation may be performed based on the initial missing pattern vector. For example, one or more frequency components of the received signal and their corresponding amplitudes may be estimated by estimating a sparse vector (e.g., the first sparse vector) . The sparse vector may be estimated by solving an optimization problem. In some embodiments, the frequency estimation may be conducted based on equation (20) .

At step 860, a missing pattern may be estimated based on the frequency estimation. For example, the missing pattern may be estimated by solving an optimization problem based on one or more previously estimated frequency components and/or their amplitudes. In some embodiments, the missing pattern may be estimated by estimating a sparse vector (e.g., the third sparse vector) . In some embodiments, the sparse vector may be estimated by solving an optimization problem (e.g., the optimization problem expressed in equation (20) ) based on one or more frequencies and/or amplitudes of the frequencies estimated at 850.

At step 870, a determination may be made as to whether the joint estimation of frequency and missing pattern is completed. This determination can be made based on any suitable criterion and/or combinations of criteria. For example, process 800 can determine that the joint estimation is completed in response to determining that a difference between the signal (e.g., a speech signal, an image, etc. ) reconstructed based on the estimation from the current iteration and the previous iteration is below a certain threshold. As another example, process 800 can determine that the joint estimation is completed when a threshold number of iterations have been executed (e.g., two, three, four, or any other suitable number) . As still another example, process 800 can determine that the joint estimation is completed in response to determining that the signal reconstructed based on the estimation from the current iteration, or one or more portions of the signal converges at a specific point. In some embodiments, the determination may be made based on data provided by the system 100, one or more user inputs, and/or any other information.

In some embodiments, in response to determining that the joint estimation is completed, process 800 may proceed to step 880 and may reconstruct the received signal. For example, missing data samples of the received signal may be recovered based on the missing pattern estimation (e.g., the missing pattern vector) . As another example, one or more frequency components and/or their amplitudes may be recovered based on the frequency estimation (e.g., the first sparse vector) and the signal received at step 820.

Alternatively, in response to determining that the joint estimation is not completed, process 800 may loop back to step 850. For example, another frequency estimation may be conducted based on the missing pattern and/or missing pattern vector calculated at step 860 (e.g., during a previous iteration of process 800) . One or more frequency components of the received signal and/or their amplitudes may be updated based on the missing pattern vector calculated at step 860 (e.g., estimating an updated sparse vector representative of sparsity of the received signal (e.g., a fourth sparse vector) ) . In some embodiments, the updated sparse vector may be estimated by solving an optimization problem based on equation (20) .

In some embodiments, the received signal may include media information, such as an image, a video, the like, or a combination thereof. The missing patterns relating to the media information may include covered or blurring portions in the picture. In some embodiments, the missing pattern may relate to spurious cell values in the media information. For example, a spurious cell value may be much brighter or darker than its surroundings. The media information may be modeled in the same manner with the real-time communication information. For example, one or portions of the media information (e.g., the noise-free part (or referred to herein as “desired part” ) , may be represented as superposition of multiple sinusoids. One or more portions of the media information (e.g., the part relating to noise) may be represented in the form of Gaussian distribution or impulsive noise distribution. In some embodiments, the media information may be modeled in a different way from the real-time communication information. For example, part of the media information, for example, the noise-free part, may be represented as superposition of multiple wavelets, and part of the media information, for example, the part relating to noise, may be represented in the form of Gaussian distribution, or impulsive noise distribution. In some embodiments, part of the media information may be represented as superposition of different types of forms, including sinusoids, wavelets, etc. Similar to the process 800, one or more iterations may be performed during the frequency and/or wavelets estimation and missing pattern estimation. During the iteration, a missing pattern may be initialized. Then, the frequency and/or wavelet estimation may be conducted. A missing pattern may be estimated based on the frequency and/or wavelet estimation calculated in the previous step. The iteration may be complete when certain conditions are satisfied. As described elsewhere in the disclosure, the certain conditions may include determining the difference between the images (that may be) reconstructed based on the estimation from the current iteration and the previous iteration, or a fixed number of iterations determined or preset by a user.

It shall be noted that the above description of the flowchart of signal processing is provided for the purpose of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, the variations and modifications may not depart the protecting scope of the present disclosure. For example, the flow diagrams may be executed or performed in any order or sequence not limited to the order and sequence shown and described in the figure. A signal model may be established when, or after a signal is received. The signal model may further be selectively established based on the received signal. In another example, at step 840, instead of initializing the missing pattern vector, the iteration may start with the initialization of a frequency estimation or wavelet estimation. Then, the missing pattern estimation may be conducted based on the initialization at the previous step. A frequency estimation or wavelet estimation may be re-conducted thereafter. The number of the iteration may be determined by, for example, the difference of the missing pattern, frequency estimation, or wavelet estimation calculated between successive iterations, or any other stopping conditions defined by the user.

FIG. 9 depicts a flowchart illustrating an example 900 of a process for performing joint frequency estimation, noise estimation and missing pattern estimation according to some embodiments of the present disclosure. The method 900 may be performed by processing logic that comprises hardware, software, or a combination thereof. In some embodiments, one or more operations of the method 900 can be performed by one or more computing devices (e.g., one or more computing devices as described below in connection with FIG. 10) executing parameter estimator 160 and/or signal recovery unit 180 of FIG. 1.

At step 910, a signal model may be constructed. For example, the signal model may represent a signal with one or more missing data samples. The signal may be modeled based on a signal without missing data and a sparse vector representative of one or more missing patterns (also referred to herein as the “third sparse vector” ) . For example, the signal may be modeled as a Schur product of the signal without missing data and the third sparse vector. In some embodiments, the signal model may be constructed based on equation (17) . The signal without missing data may be modeled as a combination of a desired signal component, a noise component, and/or any other component. In some embodiments, the signal without missing data may be modeled using a signal model constructed at 710 of FIG. 7 above.

At step 920, process 900 can receive a signal. In some embodiments, the received signal may be real-time communication information including a voice message, a video, an audio, or the like, or a combination thereof. The media information may include a picture, a frame of a video, or the like, or a combination thereof. The missing patterns relating to the real-time communication information may be caused during the transmission of the signal through a certain communication channel. In some embodiments, the signal can be received as described above in connection with step 620 of FIG. 6.

At step 930, process 900 can perform joint frequency estimation, noise estimation and missing pattern estimation based on the signal model. The frequency estimation may be conducted to calculate the existing frequency components of the received signal (e.g., frequencies and their amplitudes) , the noise estimation may be conducted to calculate the noise signal of the received signal, and the missing pattern estimation may be conducted to indicate the missing samples of the received signal. For example, the joint frequency estimation, noise estimation and missing pattern estimation may be performed by estimating the first spares vector, the second sparse vector, the residual vector, and/or the third sparse vector. The first sparse vector, the second sparse vector, the residual vector, and/or the third sparse vector may be estimated by solving one or more optimization problems. For example, the first sparse vector, the second sparse vector, and the third sparse vector may be estimated to minimize a combination of a norm of the first sparse vector, a norm of the second sparse vector, a norm of the residual vector, and/or a norm of the third sparse vector. In a more particular example, the joint frequency estimation, the noise estimation and the missing pattern estimation may be performed by solving the following convex optimization problem:

minimize ||x||₁+υs||₁+τn||₁

where vector x denotes the first sparse vector indicative of sparsity of the received signal； n denotes the second sparse vector representative of one or more noise signal； s denotes the third sparse vector representative of one or more missing patterns； υ and τ are regularization constants； and ε is a small parameter that may be predefined based on user inputs, training data, and/or any other information.

In some embodiments, the joint estimation may be performed by iteratively performing the frequency estimation, noise estimation, and missing pattern estimation. For example, during each iteration, the joint estimation may be performed by solving a convex optimization problem with response to sparse vector x, sparse vector s, and sparse vector n. In some embodiments, the joint estimation may be performed by iteratively performing one or more operations in blocks 940-970.

At step 940, a missing pattern vector (e.g., the third sparse vector s) may be initialized. For example, an initial missing pattern vector may be determined. As described in conjunction with block 840 of FIG. 8, the initial missing pattern vector x may be determined based on data stored in a storage device (e.g., the data storage device 130 or any other storage device) , one or more user inputs, and/or any other information. In some embodiments, the initial missing pattern vector may be a random vector (e.g., a vector randomly selected by, for example, the processor 120 in the system 100) . In some embodiments, the initial vector may be a predetermined vector that is preset by a user or a device.

At step 950, frequency estimation and the noise estimation may be jointly conducted. For example, the joint frequency and noise estimation may be performed by estimating the first spares vector, the second sparse vector and/or the residual vector. The first sparse vector, the second sparse vector and/or the residual vector may be estimated by solving one or more optimization problems. For example, a value of the first sparse vector and a value of the second sparse vector may be estimated to minimize a combination of a norm of the first sparse vector, and a norm of the second sparse vector. In a more particular example, the joint frequency estimation, noise estimation may be performed by solving following convex optimization problem:

minimize ||x||₁+τ||n||₁

where

indicates the initialized/estimated value of the missing pattern.

At step 960, missing pattern estimation may be performed based on the joint frequency and noise estimation. For example, a missing pattern vector may be estimated by solving one or more optimization problems based on one or more previously estimated values of the first sparse vector and the second sparse vector. In some embodiments, the third sparse vector may be estimated by solving an optimization problem (e.g., the optimization problem expressed in equation (20) ) based on one or more previously estimated values of x and n. In a more particular example, the missing pattern estimation may be performed by solving the following convex optimization problem:

minimize ||s||₁

where a denotes the estimated frequency, and

denotes the estimated impulsive noise calculated in the previous step.

At step 970, a determination may be made as to whether the joint frequency estimation, noise estimation, and/or missing pattern estimation is completed. This determination can be made based on any suitable criterion and/or combinations of criteria. For example, process 900 can determine that the joint estimation is completed in response to determining that a difference between signals (e.g., images) reconstructed based on the estimation from the current iteration and the previous iteration is below a certain threshold. As another example, process 900 can determine that the joint estimation is completed when a threshold number of iterations have been executed (e.g., two, three, four, or any other suitable number) . As still another example, process 900 can determine that the joint estimation is completed in response to determining that the signal reconstructed based on the estimation from the current iteration, or one or more portions of the signal converges at a specific point. In some embodiments, the determination may be made based on data provided by the system 100, one or more user inputs, and/or any other information.

In some embodiments, in response to determining that the joint estimation is completed, process 800 may proceed to step 880 and may reconstruct the received signal. For example, the missing patterns in the signal may be recovered based on the missing pattern estimation, noise estimation and the signal received at 920. As another example, the estimated noise signal may be removed from the signal reconstructed based on the frequency estimation, noise estimation and missing pattern estimation.

Alternatively, in response to determining that the joint estimation is not completed, process 900 may loop back to step 950. For example, another joint frequency estimation and noise estimation may be conducted based on the missing pattern calculated at step 960 (e.g., during the previous iteration of process 900) . One or more frequency components of the signal updated based on the missing pattern and/or missing pattern vector calculated at step 960 and their corresponding amplitudes may be estimated by estimating another sparse vector. In some embodiments, an updated sparse vector representative of sparsity of the received signal and/or an updated sparse vector representative of the noise component may be generated based on the missing pattern data obtained during the previous iteration of process 900.

In some embodiments, the received signal may be media information, such as an image, a video, or the like, or a combination thereof. The frequency estimation at step 930 may be substituted by wavelet estimation in the case when the signal model include superposition of multiple wavelets, or the combination of wavelet estimation and frequency estimation in the case when the signal model include superposition of multiples wavelets and sinusoids.

It should be noted that the above steps of the flow diagrams of FIGS. 6-9 can be executed or performed in any order or sequence not limited to the order and sequence shown and described in the figures. Also, some of the above steps of the flow diagrams of FIGS. 6-9 can be executed or performed substantially simultaneously where appropriate or in parallel to reduce latency and processing times. Furthermore, it should be noted that FIGS. 6-9 are provided as examples only. At least some of the steps shown in these figures can be performed in a different order than represented, performed concurrently, or altogether omitted.

The above description of the flowchart of signal processing is provided for the purpose of illustration, not intended to limit the scope of the present disclosure. For persons having ordinary skills in the art, various variations and modifications may be conducted under the teaching of the present disclosure. However, the variations and modifications may not depart the protecting scope of the present disclosure. For example, the flow diagrams may be executed or performed in any order or sequence not limited to the order and sequence shown and described in the figure. A signal model may be established when, or after a signal is received. The signal model may further be selectively established based on the received signal. In another example, at step 940, instead of initializing the missing pattern vector, the iteration may start with the initialization of a frequency estimation or wavelet estimation. Then, the missing pattern estimation may be conducted based on the initialization at the previous step. A frequency estimation or wavelet estimation may be re-conducted thereafter. The number of the iteration may be determined by, for example, the difference of the missing pattern, frequency estimation, or wavelet estimation calculated between successive iterations, or any other stopping conditions defined by the user. In another example, instead of joint estimation of frequency and noise at step 950, the frequency estimation and noise estimation may be conducted separately as depicted in equation (9) and in equation (16) .

FIG. 10 depicts the architecture of a computing device that may be used to realize a specialized system implementing the present disclosure. Such a specialized system incorporating the present teaching has a functional block diagram illustration of a hardware platform that includes user interface elements. The computer may be a general purpose computer or a special purpose computer. Both may be used to implement a specialized system for the present disclosure. This computer 1000 may be used to implement any component of the signal processing as described herein. For example, the processor 120, parameter estimator 160, etc., may be implemented on a computer such as computer 1000, via its hardware, software program, firmware, or a combination thereof. Although only one such computer is shown, for convenience, the computer functions relating to the signal processing as described herein may be implemented in a distributed fashion on a number of similar platforms, to distribute the processing load.

The computer 1000 can include COM ports 1050 connected to and from a network connected thereto to facilitate data communications. The computer 1000 also includes a central processing unit (CPU) 1020, in the form of one or more processors, for executing program instructions. The exemplary computer platform includes an internal communication bus 1010, program storage and data storage of different forms, e.g., disk 1070, read only memory (ROM) 1030, or random access memory (RAM) 1040, for various data files to be processed and/or transmitted by the computer, as well as possibly program instructions to be executed by the CPU. The computer 1000 also includes an I/O component 1060, supporting input/output between the computer and other components therein such as user interface elements 1080. The computer 1000 may also receive programming and data via network communications.

Hence, aspects of the methods of the signal processing and/or other processes, as outlined above, may be embodied in programming. Program aspects of the technology may be thought of as “products” or “articles of manufacture” typically in the form of executable code and/or associated data that is carried on or embodied in a type of machine readable medium. Tangible non-transitory “storage” type media include any or all of the memory or other storage for the computers, processors, or the like, or associated modules thereof, such as various semiconductor memories, tape drives, disk drives and the like, which may provide storage at any time for the software programming.

All or portions of the software may at times be communicated through a network such as the Internet or various other telecommunication networks. Such communications, for example, may enable loading of the software from one computer or processor into another, for example, from a management server or host computer into the hardware platform (s) of a computing environment or other system implementing a computing environment or similar functionalities in connection with the signal processing. Thus, another type of media that may bear the software elements includes optical, electrical and electromagnetic waves, such as used across physical interfaces between local devices, through wired and optical landline networks and over various air-links. The physical elements that carry such waves, such as wired or wireless links, optical links or the like, also may be considered as media bearing the software. As used herein, unless restricted to tangible “storage” media, terms such as computer or machine “readable medium” refer to any medium that participates in providing instructions to a processor for execution.

Hence, a machine-readable medium may take many forms, including but not limited to, a tangible storage medium, a carrier wave medium or physical transmission medium. Non-volatile storage media include, for example, optical or magnetic disks, such as any of the storage devices in any computer (s) or the like, which may be used to implement the system or any of its components as shown in the drawings. Volatile storage media include dynamic memory, such as a main memory of such a computer platform. Tangible transmission media may include coaxial cables； copper wire and fiber optics, including the wires that form a bus within a computer system. Carrier-wave transmission media may take the form of electric or electromagnetic signals, or acoustic or light waves such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media therefore include for example: a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD or DVD-ROM, any other optical medium, punch cards paper tape, any other physical storage medium with patterns of holes, a RAM, a PROM and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave transporting data or instructions, cables or links transporting such a carrier wave, or any other medium from which a computer may read programming code and/or data. Many of these forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a physical processor for execution.

Those skilled in the art will recognize that the present disclosure are amenable to a variety of modifications and/or enhancements. For example, although the implementation of various components described above may be embodied in a hardware device, it may also be implemented as a software only solution—e.g., an installation on an existing server. In addition, the signal processing system as disclosed herein may be implemented as a firmware, firmware/software combination, firmware/hardware combination, or a hardware/firmware/software combination.

The following examples are provided for illustration purposes, and not intended to limit the scope of the present disclosure.

The speech signal may record a normal conversation of, for example, online chat, video conferencing, etc. In some examples, the speech signal may be interrupted by keyboard clicking, which may be simulated as impulsive noise. As shown in FIG. 11, the signal is sampled at the sampling frequency of 8 KHz and the duration of signal is about 4s. In the pure noise, the clicking sound may be strong compared to the normal conversation, indicating that the speech signal may be overwhelmed by the noise. The Fourier coefficients relating to different frequency components of the signal may concentrate on the low frequency region. FIG. 12 illustrates a signal reconstructed under an exemplary joint estimation according to some embodiments of the present disclosure. As shown, a small portion of speech signal is utilized to demonstrate the reconstructed signal. At the location of approximately 200 of the time sample, the speech is corrupted by the strong noise. After the reconstruction of joint estimation of frequency and noise, the recovered speech is click-free and the estimated noise matches the one in the speech signal. As illustrated in FIG. 13, the joint estimation approach offers robust results compared to the non-joint estimation method.

Due to bad link of communications or failure of the receiver, sometimes data may be missed by the receiver. Considering the influence of data missing on the reconstruction performance, FIG. 14 illustrates a signal reconstructed under another exemplary joint estimation according to some embodiments of the present disclosure. During the test, a portion (e.g., about 40％) of the data is randomly dropped. As shown, the signal can be recovered well that the missing data and the impulsive noise can be extracted.

Having thus described the basic concepts, it may be rather apparent to those skilled in the art after reading this detailed disclosure that the foregoing detailed disclosure is intended to be presented by way of example only and is not limiting. Various alterations, improvements, and modifications may occur and are intended to those skilled in the art, though not expressly stated herein. These alterations, improvements, and modifications are intended to be suggested by this disclosure, and are within the spirit and scope of the exemplary embodiments of this disclosure.

Moreover, certain terminology has been used to describe embodiments of the present disclosure. For example, the terms “one embodiment, ” “an embodiment, ” and/or “some embodiments” mean that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present disclosure. Therefore, it is emphasized and should be appreciated that two or more references to “an embodiment” or “one embodiment” or “an alternative embodiment” in various portions of this specification are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures or characteristics may be combined as suitable in one or more embodiments of the present disclosure. In addition, the term “logic” is representative of hardware, firmware, software (or any combination thereof) to perform one or more functions. For instance, examples of “hardware” include, but are not limited to, an integrated circuit, a finite state machine, or even combinatorial logic. The integrated circuit may take the form of a processor such as a microprocessor, an application specific integrated circuit, a digital signal processor, a micro-controller, or the like.

Further, it will be appreciated by one skilled in the art, aspects of the present disclosure may be illustrated and described herein in any of a number of patentable classes or context including any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof. Accordingly, aspects of the present disclosure may be implemented entirely hardware, entirely software (including firmware, resident software, micro-code, etc. ) or combining software and hardware implementation that may all generally be referred to herein as a “circuit, ” “unit, ” “module, ” “component, ” or “system. ” Furthermore, aspects of the present disclosure may take the form of a computer program product embodied in one or more computer readable media having computer readable program code embodied thereon.

A computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that may communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device. Program code embodied on a computer readable signal medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber cable, RF, etc., or any suitable combination of the foregoing.

Computer program code for carrying out operations for aspects of the present disclosure may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Scala, Smalltalk, Eiffel, JADE, Emerald, C++, C#, VB. NET, Python or the like, procedural programming languages, such as the “C” programming language, Visual Basic, Fortran 2003, Perl, COBOL 2002, PHP, ABAP, dynamic programming languages such as Python, Ruby and Groovy, or other programming languages. The program code may execute entirely on the user’s computer, partly on the user’s computer, as a stand-alone software package, partly on the user’s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user’s computer through any type of network, including a local area network (LAN) or a wide area network (WAN) , or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider) or in a cloud computing environment or offered as a service such as a Software as a Service (SaaS) .

Furthermore, the recited order of processing elements or sequences, or the use of numbers, letters, or other designations therefore, is not intended to limit the claimed processes and methods to any order except as may be specified in the claims. Although the above disclosure discusses through various examples what is currently considered to be a variety of useful embodiments of the disclosure, it is to be understood that such detail is solely for that purpose, and that the appended claims are not limited to the disclosed embodiments, but, on the contrary, are intended to cover modifications and equivalent arrangements that are within the spirit and scope of the disclosed embodiments. For example, although the implementation of various components described above may be embodied in a hardware device, it may also be implemented as a software only solution—e.g., an installation on an existing server or mobile device. In addition, the financial management system disclosed herein may be implemented as a firmware, firmware/software combination, firmware/hardware combination, or a hardware/firmware/software combination.

Similarly, it should be appreciated that in the foregoing description of embodiments of the present disclosure, various features are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure aiding in the understanding of one or more of the various inventive embodiments. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed subject matter requires more features than are expressly recited in each claim. Rather, inventive embodiments lie in less than all features of a single foregoing disclosed embodiment.

Claims

A method for processing a signal， the method comprising：

constructing a signal model representative of the signal；

determining an initial missing pattern vector relating to the signal model；

conducting， by a processor， a frequency estimation on the signal based on the initial missing pattern vector；

estimating a missing pattern based on the frequency estimation； and

reconstructing the signal based on the frequency estimation and the missing pattern.
The method of claim 1， wherein conducting the frequency estimation comprises estimating a first sparse vector representative of sparsity of the signal.
The method of claim 2， wherein at least one component of the first sparse vector corresponds to a frequency component of the signal.
The method of claim 2， wherein estimating the missing pattern comprises estimating an updated missing pattern vector based on the first sparse vector.
The method of claim 2， further comprising conducting a noise estimation based on the first sparse vector.
The method of claim 5， wherein conducting the noise estimation comprises estimating a second sparse vector based on the first sparse vector， wherein the second sparse vector corresponds to impulsive noise.
The method of claim 1， wherein conducting the missing pattern estimation comprises estimating a third sparse vector.
The method of claim 1， wherein at least a component of the initial missing pattern vector is zero.
The method of claim 1， wherein the signal model comprises a superposition of a plurality of sinusoids， and wherein each of the sinusoid corresponds to a frequency component of the signal.
The method of claim 1， wherein the missing pattern corresponds to at least a portion of an image.
A system for processing a signal， the system comprising：

at least one hardware processor that is configured to：

construct a signal model representative of the signal；

determine an initial missing pattern vector relating to the signal model；

conduct a frequency estimation on the signal based on the initial missing pattern vector；

estimate a missing pattern based on the frequency estimation； and

reconstruct the signal based on the frequency estimation and the missing pattern.
The system of claim 11， wherein， to conduct the frequency estimation， the hardware processor is further to estimate a first spare vector representative of sparsity of the signal.
The system of claim 12， wherein at least one component of the first sparse vector corresponds to a frequency component of the signal.
The system of claim 12， wherein the hardware processor is further configured to estimate the missing pattern comprising estimating an updated missing pattern vector based on the first sparse vector.
The system of claim 12， wherein the hardware processor is further configured to conduct a noise estimation based on the first sparse vector.
The system of claim 15， wherein to conduct the noise estimation， the hardware processor is further to estimate a second sparse vector based on the first sparse vector， wherein the second sparse vector corresponds to impulsive noise.
The system of claim 11， wherein， to conduct the missing pattern estimation， the hardware processor is further to estimate a third sparse vector.
The system of claim 11， wherein at least a component of the initial missing pattern vector is zero.
The system of claim 11， wherein the signal model comprises a superposition of a plurality of sinusoids， and each of the sinusoid corresponds to a frequency component of the signal.
The system of claim 11， wherein the missing pattern corresponds to at least a portion of an image.
A non-transitory computer-readable medium containing computer-executable instructions that， when executed by a processor， cause the processor to perform a method for processing a signal， the method comprising：

constructing a signal model representative of the signal；

determining an initial missing pattern vector relating to the signal model；

conducting a frequency estimation on the signal based on the initial missing pattern vector；

estimating a missing pattern based on the frequency estimation； and

reconstructing the signal based on the frequency estimation and the missing pattern.
The non-transitory computer-readable medium of claim 21， wherein the method further comprises estimating a first spare vector representative of sparsity of the signal.
The non-transitory computer-readable medium of claim 22， wherein at least one component of the first sparse vector corresponds to a frequency component of the signal.
The non-transitory computer-readable medium of claim 22， wherein the method further comprises estimating the missing pattern comprising estimating an updated missing pattern vector.
The non-transitory computer-readable medium of claim 22， wherein the method further comprises conducting a noise estimation based on the first sparse vector.
The non-transitory computer-readable medium of claim 25， wherein conducting the noise estimation comprises estimating a second sparse vector based on the first sparse vector， wherein the second sparse vector corresponds to impulsive noise.
The non-transitory computer-readable medium of claim 21， wherein conducting the missing pattern estimation comprising estimating a third sparse vector.
The non-transitory computer-readable medium of claim 21， wherein at least a component of the initial missing pattern vector is zero.
The non-transitory computer-readable medium of claim 21， wherein the signal model comprises a superposition of a plurality of sinusoids， and wherein each of the sinusoid corresponds to a frequency component of the signal.
The non-transitory computer-readable medium of claim 21， wherein the missing pattern corresponds to at least a portion of an image.